#34205187

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #12987

Subgroups

	subgroup	G/H
center	C₈	#14579133
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2955	C₂
socle	C₂×C₇₉	#1869
prime radical	#575×C₇₉	C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2955	D₇₉
hypocenter	C₇₉	#12987
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2955

456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 143
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 143 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,83)(81,89)(82,94)(84,103)(85,106)(86,100)(87,99)(88,110)(90,116)(91,118)(92,121)(93,98)(95,127)(96,126)(97,129)(101,108)(102,124)(104,132)(105,125)(107,114)(109,115)(111,130)(112,139)(113,136)(117,123)(119,131)(120,137)(122,134)(128,142)(133,135)(138,141)(140,143);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,82,85,105,86,96,107,133,87,95,108,97,84,102,104,128)(81,88,109,112,93,111,119,138,90,113,92,120,91,117,122,140)(83,94,106,125,100,126,114,135,99,127,101,129,103,124,132,142)(89,110,115,139,98,130,131,141,116,136,121,137,118,123,134,143);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81,85,92,86,93,107,122,87,90,108,109,84,91,104,119)(82,88,97,112,96,111,128,138,95,113,105,120,102,117,133,140)(83,98,101,131,100,116,132,121,99,118,106,134,103,89,114,115)(94,123,125,141,126,110,135,137,127,130,129,143,124,136,142,139)

Polycyclic presentation:

2^8,79|a^2=f,b^2=df,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,i^a=i^78,d^b=dh,e^b=eh,f^b=fh,i^b=i^78,d^c=dh,e^c=eh,f^c=fh

Conjugacy classes & character table

1780 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 86486);